## Engage NY Eureka Math 8th Grade Module 7 Lesson 14 Answer Key

### Eureka Math Grade 8 Module 7 Lesson 14 Exercise Answer Key

Opening Exercise
a. Write an equation for the area, A, of the circle shown.

A < π(6.3) 2
< 39.69π
The area of the circle is 39.69π cm 2 .

b. Write an equation for the circumference, C, of the circle shown.

C < 2π(9.7)
< 19.4π
The circumference of the circle is 19.4π mm.

c. Each of the squares in the grid below has an area of 1 unit 2 .

i. Estimate the area of the circle shown by counting squares.
Estimates will vary. The approximate area of the circle is 78 units 2 .

ii. Calculate the area of the circle using a radius of 5 units. Use 3.14 as an approximation for π.
A≈3.14(5) 2
≈78.5
The area of the circle is approximately 78.5 units 2 .

Exercises 1–4

Exercise 1.
Gerald and Sarah are building a wheel with a radius of 6.5 cm and are trying to determine the circumference. Gerald says, “Because 6.5 × 2 × 3.14 < 40.82, the circumference is 40.82 cm.” Sarah says, “Because
6.5 × 2 × 3.10 < 40.3 and 6.5 × 2 × 3.21 < 41.73, the circumference is somewhere between 40.3 and 41.73.” Explain the thinking of each student.
Gerald is using a common approximation for the number π to determine the circumference of the wheel. That is why he used 3.14 in his calculation. Sarah is using an interval between which the value of π falls, based on the work we did in class. We know that 3.10 < π < 3.21; therefore, her calculations of the circumference uses numbers we know π to be between.

Exercise 2.
Estimate the value of the number (6.12486…) 2 .
6.12486 2 < (6.12486…) 2 < 6.12487 2
37.5139100196 < (6.12486…) 2 < 37.5140325169
(6.12486…) 2 < 37.51 is correct up to two decimal digits.

Exercise 3.
Estimate the value of the number (9.204107…) < sup>2 < /sup>.
9.204107 2 < (9.204107… ) 2 < 9.204108 2
84.715585667449 < (9.204107…) 2 < 84.715604075664
(9.20410…) 2 < 84.715 is correct up to three decimal digits.

Exercise 4.
Estimate the value of the number (4.014325…) < sup>2 < /sup>.
4.014325 2 < (4.014325…) 2 < 4.014326 2
16.114805205625 < (4.014325…) 2 < 16.114813234276
(4.014325…) 2 < 16.1148 is correct up to four decimal digits.

### Eureka Math Grade 8 Module 7 Lesson 14 Problem Set Answer Key

Question 1.
Caitlin estimated π to be 3.10 < π < 3.21. If she uses this approximation of π to determine the area of a circle with a radius of 5 cm, what could the area be?
The area of the circle with radius 5 cm will be between 77.5 cm2 and 80.25 cm2.

Question 2.
Myka estimated the circumference of a circle with a radius of 4.5 in. to be 28.44 in. What approximate value of π did she use? Is it an acceptable approximation of π? Explain.
C < 2πr
28.44 < 2π(4.5)
28.44 < 9π
$$\frac{28.44}{9}$$ < π
3.16 < π
Myka used 3.16 to approximate π. Student responses may vary with respect to whether or not 3.16 is an acceptable approximation for π. Accept any reasonable explanation.

Question 3.
A length of ribbon is being cut to decorate a cylindrical cookie jar. The ribbon must be cut to a length that stretches the length of the circumference of the jar. There is only enough ribbon to make one cut. When approximating π to calculate the circumference of the jar, which number in the interval 3.10 < π < 3.21 should be used? Explain.
In order to make sure the ribbon is long enough, we should use an estimate of π that is closer to 3.21. We know that 3.10 is a fair estimate of π but less than the actual value of π. Similarly, we know that 3.21 is a fair estimate of π but greater than the actual value of π. Since we can only make one cut, we should cut the ribbon so that there is a little more than we need, not less than. For that reason, an approximation of π closer to 3.21 should be used.

Question 4.
Estimate the value of the number (1.86211…)2.
1.862112 < (1.86211…)2 < 1.862122
3.4674536521 < (1.86211…)2 < 3.4674908944
(1.86211…)2 < 3.4674 is correct up to four decimal digits.

Question 5.
Estimate the value of the number (5.9035687…)2.
5.90356872 < (5.9035687…)2 < 5.90356882
34.85212339561969 < (5.9035687…)2 < 34.85212457633344
(5.9035687…)2 < 34.85212 is correct up to five decimal digits.

Question 6.
Estimate the value of the number (12.30791…)2.
12.307912 < (12.30791…)2 < 12.307922
151.4846485681 < (12.30791…)2 < 151.4848947264
(12.30791…)2 < 151.484 is correct up to three decimal digits.

Question 7.
Estimate the value of the number (0.6289731…)2.
0.62897312 < (0.6289731…)2 < 0.62897322
0.39560716052361 < (0.6289731…)2 < 0.39560728631824
(0.6289731…)2 < 0.395607 is correct up to six decimal digits.

Question 8.
Estimate the value of the number (1.112223333…)2.
1.1122233332 < (1.112223333…)2 < 1.1122233342
1.2370407424696289 < (1.112223333…)2 < 1.2370407446940756
(1.112223333…)2 < 1.23704074 is correct up to eight decimal digits.

Question 9.
Which number is a better estimate for π, $$\frac{22}{7}$$ or 3.14? Explain.
Allow for both answers to be correct as long as the student provides a reasonable explanation.
A sample answer might be as follows.
I think that $$\frac{22}{7}$$ is a better estimate because when I find the decimal expansion, $$\frac{22}{7}$$≈3.142857…; compared to the number 3.14, $$\frac{22}{7}$$ is closer to the actual value of π.

Question 10.
To how many decimal digits can you correctly estimate the value of the number (4.56789012…)2?